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CHAPTER 2

MICROWORLD MODELING: IMPACT OFLIQUID AND ROUGHNESSPIERRE LAMBERT and STEPHANE REGNIER

2.1 INTRODUCTION

This chapter is a complement to the physics introduced in the first chapter. Moreparticularly, it deals with the analysis of the impact of liquid and of roughnesson the behavior of the microobjects.

Liquid environments have been tremendously studied these last years becausethey are supposed to suppress capillary forces since there is no liquid–vaporinterface anymore because this environment reduces the electrostatic effect dueto higher dielectric constants and so on. These environments nevertheless implyspecific phenomena such as the double-layer forces and larger hydrodynamicforces. It is shown how surface forces are actually affected by the liquid environ-ment, including a microscopic analysis based on AFM measurements. Finally, wepropose to address the issue of the surface topography and the roughness descrip-tion (e.g., statistical description versus a fractal one to model the experimentalroughness profile).

2.2 LIQUID ENVIRONMENTS

2.2.1 Classical Models

2.2.1.1 Double-Layer ForcesElectrostatic interactions are due to the presence of fixed or induced charges onthe surface of particles. In the polar regions (e.g., in liquid environments), the

Robotic Microassembly, edited by Michael Gauthier and Stephane RegnierCopyright © 2010 the Institute of Electrical and Electronics Engineers, Inc.

55

56 MICROWORLD MODELING: IMPACT OF LIQUID AND ROUGHNESS

main electrical charge mechanisms are:

• The dissociation of chemical surface groups• The adsorption process

The surface ions attract opposite charges contained in the liquid medium. Theyform an area of counterions around the interface and ensure global electronicneutrality. The length of this characteristic phenomenon, called Debye length,depends on the concentration of ions. In the particular case of liquid colloidalsolutions,1 the concentration usually exceeds 10−7 mol/L, so the Debye lengthis similar to the size of the particles. In this case, an electric double layer (EDL)is formed, consisting of the surface charges and countercharges. EDL affectsmost dynamic phenomena in colloidal suspensions, as well as their stability.Indeed, the EDL creates a repulsion between two charged colloids, which wouldaggregate because of van der Waals forces. The van der Waals attraction andEDL repulsion form the basis of the famous DLVO theory on colloidal stability[60]. The DLVO theory is composed of two forces; repulsion due to the electricdouble layer and attraction due to the van der Waals force.

No comprehensive theory for the quantitative description of EDL has yet beenpublished. Currently, two main approaches exist [1]:

• A statistical thermodynamic approach, the mathematical complexity ofwhich prevents its implementation in practice

• A phenomenological approach based on a local thermodynamic equilib-rium, used in the DLVO theory, but which does not take into account thecorrelations between ions, adielectric saturation, and the finite size of ions

2.2.1.2 Qualitative Models of the Electric Double LayerEDL models have been designed to describe the change in electric potentialat the solid–liquid interface. Several qualitative EDL models exist, differingmainly on how to visualize the spatial distribution of the countercharge. Theyare, in chronological order, the Helmholtz model (1879), the Gouy–Chapmanmodel (1913), the Stern model (1924), and the Gouy–Chapman–Stern–Grahamemodel (GCSG) (1947) also called triple layer. First, we will examine how thelatter explains EDL. Then by successive approximations, we examine how theolder models perform.

The GCSG model breaks down the solid–liquid interface into three layers ofcharge (see Fig. 2.1):

• The first layer: The surface layer where the ions are adsorbed, determiningthe potential of solids (e.g., H+ and OH− for an oxide metal such as SiO).The charge σ0 and potential ψ0 are associated with this layer.

1Liquid containing suspended particles the size of which is between 1 nm and 1 μm.

LIQUID ENVIRONMENTS 57

Distance0

yb

s0 sb

yd

Hydrated ion

Adsorbed ions

H2O

Ca2+

PO43−

sd

PEHPIH

Surfacelayer

Sternlayer

Diffuse layer Solution

Debye length k−1

y0

Figure 2.1. Triple layer. Equations are: σ0 + σβ + σd = 0, ψ0 − ψβ = σ0/C1, and ψβ −ψd = (σ0 + σβ)/C2 = −σd/C2. The example shown is the case of a surface covered withcalcium phosphate [58].

• The second layer: The compact layer of dehydrated ions interacting stronglywith the surface (ions specifically adsorbed). The center of these ions islocated at the internal Helmholtz plane (IHP). With this layer, the chargeσβ and the potential ψβ are defined.

• The third layer: In this layer, the diffuse layer of ions and hydrated coun-terions are slightly attracted. The plane where the diffuse layer begins iscalled the external Helmholtz plan (EHP). It combines the charge σd andthe potential ψd .


This model has been designed to take into account the absorption of ionson the surface of inorganic oxides, where the surface charge depends on thepH, dissociation constants, and the adsorption reactions. From the point of viewof electrical engineering, separation of charges between the different planes iscomparable to three capacities in series, giving a total capacity:

1

Ct

= 1

C1+ 1

C2+ 1

Cd

(2.1)

with σ0 = C1(ψ0 − ψβ) and σβ = C2(ψβ − ψd).

2.2.1.3 Stern ModelIf C2 is neglected (i.e., neglecting the finite size of the second layer), thenψβ = ψd and 1/Ct = 1/C1 + 1/Cd . Adamczyk [1] notes that for interactionsof particles, the Stern layer’s role is negligible except for very short separationsaround 5–10 A.

Two limit cases of this model may arise:

• If the potential and ionic strength are low (n0 < 0.01 mol/L), then C1 � Cd

and the model is reduced to the diffuse layer model (Gouy–Chapman model)with 1/Ct = 1/Cd .

• If the potential and ionic strength are high, then the model is reduced to theconstant capacity model (Helmholtz model) with 1/Ct = 1/C1.

The model used in this book and these circumstances is the Gouy–Chapmanmodel.

2.2.1.4 Gouy–Chapman Electric Double-Layer ModelA flat surface is considered. The charge on that surface influences the ion distri-bution in nearby layers of the electrolyte. The electrostatic potential, ψ , and thevolume charge density, ρ, which is the excess of charges of one type over theother, are related by the Poisson equation:

∇2ψ = − ρ

ε0εr

(2.2)

with εr relative permitivitty (or dielectric constant) of electrolyte.The ion distribution in the charged surface region is determined by (i) temper-

ature and (ii) the energy required, wi , to bring the ion from an infinite distanceaway (where ψ = 0) to the region where the electrostatic potential is ψ . Thisdistribution is given by a Boltzmann equation:

ni = n0i e

−wi/(kBT ) = n0i e

−zi eψ/(kBT ) (2.3)

where n0i is the number of ions of type i per unit volume of bulk solution,

wi = zieψ , the energy expended in bringing an ion from an infinite distance


from the surface to a point where the potential is ψ and zi is the valency of ionspecies i.

The volume charge density at ψ is

ρ =∑

i

niziee−zi eψ/(kBT ) (2.4)

Thus the combination of this with the Poisson equation gives thePoisson–Boltzmann equation:

∇2ψ = −∑

i niziee−zieψ/(kBT )

ε0εr

(2.5)

2.2.1.5 Zeta PotentialWhen an electric field is applied in a suspension containing charged particles,the particles acquire a degree of mobility depending on their charges. Withinthe double layer, there is a plane delineating the ions from the particle in thismovement. The plane, called shear plane, is defined on the basis of hydrodynamicconsiderations.

However, it is possible to link this plane to the chemical description of thedouble layer. Indeed, it is usually very close to the plane delineating the compactlayer from the diffuse layer—the external Helmoltz plane.

The electric potential in the shear plane is commonly noted as the zetapotential ζ . It is easily accessible by electrokinetic measures (electrophoresis,acoustophoresis, flow potential, etc.). See Hunter [27] for more details. It isoften used as a potential surface in the form of an electrostatic interaction.

2.2.1.6 Poisson–Boltzmann EquationThe Poisson–Boltzmann (PB) equation needed to solve the problem is stronglynonlinear due to the exponential terms, which precludes an analytical solution.Some solutions are available in the literature in the form of tables, graph solu-tions, elliptical integrals, and elliptical functions [39]. Nevertheless, two differentapproximations can be made to arrive at an analytical resolution.

For a symmetrical electrolyte composed of two types of ions, where z1 = z2 =z, Eq. 2.5 becomes

∇2ψ = − 1

ε0εr

[n0zee−zeψ/(kBT ) − n0zeezeψ/(kBT )

](2.6)

Using trigonometric relations, we find

∇2ψ = κ2 sinh ψ (2.7)


where ψ = zeψ/kT is the reduced electric potential and

κ−1 =(

ε0εkT

2e2I

)1/2

(2.8)

where κ−1 is the Debye length, I = z2n0 is the ionic force of the electolyte, n0

its volumic concentration, and z the valency.Another common form of this equation is the linear PB equation, obtained by

linearizing exponential terms, provided that max(zieψ/kT ) < 1 or max(ziψ) <

25 mV is at 25◦. This is the linear Debye–Heckel approximation:

∇2ψ = κ2ψ (2.9)

For an order of magnitude, the Debye length κ−1 in water and at room tem-perature varies between about 0,4 nm (for a solution of Na3PO4 at 0,1 mol/L)and 30 nm (for a solution of KCl at 10−6 mol/L) [1].

2.2.2 Sphere–Sphere and Sphere–Plane Interactions

We consider in this section interaction between a sphere of radius a and a planeor interaction between spheres of radius R1 and R2 with h separation distance.

There are two main approaches to calculating an approximation of the inter-actions between spheres: the method of linear superposition (LSA) restricted tolarge distances (κh>1 with h the separation distance) and Derjaguin’s approxi-mation (1940) appropriate for small distances (κh < 1) and small Debye lengths(κ−1 < a). However, Sader et al. [49] obtained a valid formula for any κh, whilekeeping a simple analytical form.

2.2.2.1 LSA MethodThe LSA method [3] postulates that the solution to the PB equation for a systemof two particles can be built as the linear superposition of the solutions for isolatedparticles. This is justified by the fact that the electric potential at distances largerthan κ−1 drops to low values and its description can be described by the linearPB equation. Therefore, the solution of this equation in this region is calculatedassuming the additivity of the potential of isolated particles:

ψs = ψs1 + ψs2 (2.10)

The solution of the linear PB equation for a sphere with a radius a and a smallpotential ψ s < 1 is

ψ = ψs

a

re−κ(r−R) (2.11)

where r is the distance from the center. However, in order to be close to thenonlinear solutions, ψs can be replaced by the effective potential Y determined by


the numerical solution of the exact PB equation for a sphere. For a symmetricalelectrolyte, Sader [48] gives a formula valid for any κa and for any ψs withψs < 200 mV:

Y = kT

ze

ψ s + 4γ�κR

1 + �κRwith � = ψ s − 4γ

2γ 3and γ = tanh(

ψ s

4)

(2.12)

For a small potential, we can calculate the force between the particles. Wethen get for h>κ−1:

Fss = 4πε0εR1R21 + κr

r2Y1Y2e

−κh (2.13)

Wss = 4πε0εR1R2

R1 + R2 + hY1Y2e

−κh (2.14)

where r = R1 + R2 + h and h is the distance between spheres.In addition, where a radius tends to infinity, Eqs. 2.13 and 2.14 are reduced

to the form describing the sphere–plane interaction:

Fsp = 4πε0εκaY1Y2e−κh (2.15)

Wsp = 4πε0εaY1Y2e−κh (2.16)

Finally, the LSA method can also be applied to the plane–plane interaction.The potential of a single plane for h >κ−1 is

ψ = Ye−κh (2.17)

where Y = 4 tanh(ψ s/4) (because κa → ∞ in Eq. 2.12 in the plane case). In thecase of an asymmetrical electrolyte, Ohshima [42] gives analytical formulas tocalculate Y . The LSA assumption gives the following terms:

� = 2ε0εκ2Y1Y2e

−κh (2.18)

Wpp = 2ε0εκY1Y2e−κh (2.19)

These two formulas can be found in Israelachvili [28]. It is noted that for asmall potential where Y ≈ ψs, Eq. 2.19 corresponds to the solution of the linearDB equation where κh � 1.

Thanks to their simple mathematical form, Eqs. 2.13–2.19 are widely used innumerical simulations for particle absorption problems. The disadvantage of theLSA method is that it becomes less effective for small separations (h < κ−1). Inthis case, Derjaguin’s approximation can be used.


2.2.2.2 Derjaguin MethodAccording to this method [14], the interactions between spheres can be calculatedas the sum of the interactions to elementary surfaces (rings) with a plane geome-try. The Derjaguin method connects the interaction energy per area unit betweentwo planes Wpp and the interaction energy between two curved surfaces Wss withthe equation (for Derjaguin, we use classical notation, z interaction distance):

Wss = 2πGD

∫ ∞

zmin

Wpp(z) dz (2.20)

where zmin is the minimum distance between the curved surfaces and GD theDerjaguin factor geometry. The latter is calculated easily for simple geometries,GD = R1R2/(R1 + R2) for two spheres, GD = R/2 for two identical spheres,and GD = R for the sphere–plane configuration.

However, the approximation assumes that the scope of the interaction energyis much shorter than the radius of curvature of the particles. This means that theinteraction energy between the two particles is created in a small region aroundzmin (which admits the limit of infinite integration in Eq. 2.20. The Derjaguinmethod is valid only if κRi � 1. In practice, the approximation remains validfor κRi > 5 [26], which corresponds to micrometer colloidal particles in an elec-trolyte diluted to approximately 10−4 or to globular proteins in an electrolytewith a physiological concentration (≈ 0, 15 mol/L).

By using Eq. 2.20, we get

Fss = 2πε0εκ

1 − e−2κzGD

[±(ψ2s1 + ψ2

s2)e−2κz + 2ψs1ψs2e

−κz]

(2.21)

Wss = πε0εGD

[∓(ψ2

s1 + ψ2s2) ln(1 − e−2κz) + 2ψs1ψs2 ln

1 + e−κz

1 − e−κz

](2.22)

where the ± sign depends on the conditions at the limit of the resolution: theplus sign for areas with constant charges (c.c.) and the minus sign for surfaceswith constant potential (c.p.).

Equation 2.2 is called the HHF formula, named after its authors Hogg, Healy,and Feurstenau [26]. When the surface potential is equal and ψs1 = ψs2 = ψ rms ,Eq. 2.22 is reduced to the form given by [14]

Wss = ∓4πε0εGDψ2s ln(1 ± e−κz) (2.23)

Equations 2.22 and 2.23 are often used in the literature to determine thestability criteria of colloidal suspensions. It may also be noted that this methodhas been generalized to convex bodies of any shape [1]. Figure 2.2 gives acomparison of the LSA model (Eqs. 2.14 and 2.16) and the linear HHF Derjaguinmodel (Eq. 2.22) with numerical solutions of the nonlinear PB equation.

However, the Derjaguin method becomes less effective for separations exceed-ing κ−1. This is because the Derjaguin approximation considers the interaction


energy between two elementary surfaces that are similar face to face, while inter-action energy between two planes is the interaction energy in a point of a planedue to any point of the other plane [4]. This constraint creates an overstatementof interactions and a false asymptotic dependence of Wss = f (z) (in the formula,it lacks dependency on 1/z when z increases).

Moreover, the surface potentials must be low. Hogg et al. [26] showed that theapproximation was good up to 50 mV (or ψ si ≤ 2). In summary, the conditionsto check for using the Hogg, Healy, and Feurstenau (HHF) formula are κz < 1,ψ si ≤ 2 and κRi > 5.

2.2.2.3 Improved FormulasSader et al. [49] have demonstrated that the HHF formula could be easilyamended to apply to any κz while retaining its analytical simplicity. They sum-marized their analysis from Bell et al. [3] for ψ si < 2, κRi > 5, and areas withconstant potential from which they obtained the amended HHF formula:

Wss = πε0εR1R2

R1 + R2 + h

[(ψ2

s1 + ψ2s2) ln(1 − e−2κz) + 2ψs1ψs2 ln

1 + e−κz

1 − e−κz

](2.24)

Figure 2.2. Reduced interaction energy W(e/kT ) sphere–sphere (low diagram) andsphere–plan (top diagram) computed for ψ s1 = 3, ψ s2 = −1.5, and κa = 5: −•−•−exact numerical evaluation for the c.c. case; −◦−◦− iexact numerical evaluation for thec.p. case; · · · · · · linear HHF model for the c.c. case; −·−·−linear HHF model for thec.p. case; — — — LSA model [1].


We note that for a small κz, the amended HHF formula coincides with theHHF formula of Eq. 2.22 and for a large κz, it is reduced to the LSA formulaof Eq. (2.14).

Finally, the same authors also offer a simple and effective formula for twoidentical spheres with moderate to high potential (ψ si ≤ 4 then ψsi ≤ 100 mV),valid for any κz and κRi > 5. Using the solution of the nonlinear PB equationfor a single sphere, they get the expressions:

Fss = 4πε0εR2

r2Y 2(z)

[ln(1 + e−κz) + κr

e−κz

1 + e−κz

](2.25)

Wss = 4πε0εR2

rY 2(z) ln(1 + e−κz) (2.26)

with Y (z) = 4eκz/2 tanh−1[e−κz/2/ tanh(

ψs4 )

]∀z and Y (z) ≈ 4 tanh(ψ s/4) for

κz > 2.

2.2.2.4 DLVO TheoryThe DLVO theory is linked to the Stern model and studies, in particular, thediffuse layer. It assumes that the total interaction between two surfaces is thesum of double-layer repulsion and the van der Waals attraction. For example, inthe case of an interaction between spheres of radius a with weak potential andh � a, the potentiel and the force are equal to (using 2.23):

Fss = −A132a

12h+ 4πε0ε

a

2ψ2

s

[ln(1 + e−κz)

]Wss = −A132a

12h2+ 4πε0ε

a

2ψ2

sκ

1 + e−κz

(2.27)

An another example is the interaction between a sphere of radius a and aplane:

Fsp = −A132a

6h+ 4πε0εaψ2

s

[ln(1 + e−κz)

](2.28)

Wsp = −A132a

6h2+ 4πε0εaψ2

sκ

1 + e−κz(2.29)

You can use these equations for micromanipulation in a liquid environment,for example. We can also use any expressions previously defined for the EDL.Contrary to the double-layer interaction, the van der Waals interaction is muchless sensitive to changes in the electrolyte’s concentration and pH. Van der Wallsinteraction can therefore be considered fixed in the first approximation. Besides, itstill exceeds the double-layer repulsion at short distances because WvdW ∝ −1/zn.Thus, according to the concentration of the electrolyte and the potential or surfacicdensity of the charges, different scenarios, schematically illustrated in Figure 2.3can occur [28]:


Energy barrierDouble-layerrepulsion

0

W

Wo

105

b

0

Inte

ract

ion

ener

gy W

Van der Waals attraction

Total

W

0

a

b

c

deIncreasing salt,decreasing surfacepotential

b

Secondary minimum (Ws)

Primary minimum (Wp)

Distance, D (nm)

Figure 2.3. Schematic profiles of interaction energy [28].

1. Surfaces repel strongly; colloids are thermodynamically stable.2. Surfaces can stabilize at a secondary minimum if this minimum is fairly

deep and the colloids remain kinetically stable.3. Surfaces stabilize at the secondary minimum; colloids are slowly bonded.

This phenomenon is called flocculation: the formation of agglomeratedparticles the size of which is sufficient to settle or to float. Simple agitationmay cancel this flocculation.

4. At the critical concentration of coagulation, surfaces can remain at thesecondary minimum or coagulate; colloids bond quickly.

5. Surfaces and colloids coagulate quickly (formation of a precipitate).

Finally, apart from lyophobe colloids, many differences2 between DLVO the-ory and experimental observations have been reported for industrial and natural

2DLVO theory predicts stability or instability contrary to experiments.


systems [25]. These differences are due to non-DLVO short-range interactions,such as hydration forces in an aqueous environment [66], steric forces,3 andhydrodynamic forces. Taking these interactions into account is also part of theextended DLVO or XDLVO approach [10, 11].

2.2.2.5 XDLVO ModelThe XDLVO model is an extended version of the previous model and includessolvation forces (or hydration forces in water). These interactions are also referredto as acid–base Lewis interactions. In water, these forces will be attractive forhydrophobic surfaces and repellent for hydrophilic surfaces. Thus, the compo-nents of the surface energy can be classified into two categories [20, 21]:

• The first gathering dispersive interactions, so-called van der Waals Lifshitzinteractions, are designated by the exhibitor LW and are mainly due to thevan der Waals effect between micro objects.

• The second gathering the interactions linked to the donor–acceptor mecha-nism of electrons; these interactions form the Lewis acid–base theory, aredesignated by the exhibitor AB [16], and are mainly due to the liquid effectadded by the DLVO theory.

This hypothesis can be written and are follows:

γ = γ AB + γ LW (2.30)

In the Lewis theory, a base is a compound carrying an electronic doublet andis thus an electron donor. An acid is a compound carrying a gap and is thus anelectron acceptor. This approach allows us to describe the acid–base propertyusing two parameters, γ + and γ −, where γ + is the acceptor electron parameter(or Lewis acid) and γ − is the electron donor parameter (or Lewis base). Thecomponent γ AB surface energy is a function of γ + and γ − according to therelationship

γ AB = 2√

γ +γ − (2.31)

The total surface energy can be written as

γ = γ AB + γ LW = γ LW + 2√

γ +γ − (2.32)

with γ LW incorporating the previous DLVO model. We can therefore write theinterfacial energy in the case of a contact between a solid and a liquid as

γSL = γ ABSL + γ LW

SL (2.33)

3Adsorption of neutral polymers by particles creates a steric repulsion due to the repulsive interactionsbetween the polymer chains.


with expressions of components proposed by Good:

γ ABSL = 2(

√γ +

S γ −S +

√γ +

L γ −L −

√γ +

S γ −L −

√γ +

L γ −S ) (2.34)

γ LWSL = (

√γ LW

S −√

γ LWL )2 (2.35)

By combining Eqs. 2.33, 2.34, and 2.35, we get

γSL = γS + γL − 2√

γ LWS γ LW

L − 2√

γ +S γ −

L − 2√

γ +L γ −

S (2.36)

This equation and the Young–Dupre relationship give the equation for deter-mining the components of the solid’s surface energy:

γL[1 + cos(θ)] = 2√

γ LWS γ LW

L + 2√

γ +S γ −

L + 2√

γ +L γ −

S (2.37)

This formula gives an equation with three unknowns (each α component of thesolid’s surface energy) that are solved by measuring the angles of contact withthree different liquids [59]. Water is the reference for determining the acid–basecomponents of other liquids:

γ +water = γ −

water = 25.5 mJ/m2 (2.38)

The term added to the DLVO formulation is due to acid–base interactions.The XDLVO theory for a force between a sphere and a plane can be written:

F = FvdW + Fedl + FAB = FLW + FAB

= −A132a

6h+ 4πε0εaψ2

s

[ln(1 + e−κz)

] − P exp(z0 − z)

λ(2.39)

with P the pull-off force previously described with 32πRWAB

132 ≤ P ≤ 2πRWAB132,

W132 = γ AB13 + γ AB

23 − γ12 (different energies can be estimated using the previousmethod), z0 the minimum equilibrium distance z0 = 0.157 nm, and λ the decaylength. This length is estimated to 0.6 nm.

The form of the double-layer equation is once again chosen according tothe assumptions described above, but for microscopic object you can use thisapproximation.


2.2.2.6 Hydrodynamic ForcesIn the microworld, the Reynolds number, which characterizes liquid flow, isusually very low (<1). The flow is thus highly laminar. This part will presentthe hydrodynamical law valid at the microscopic scale.

For example, in case of a microobject placed in a uniform liquid flow, Stokes’slaw directly gives the hydrodynamic force applied on the object. This law is validwhen the flow’s Reynolds number is lower than 1 and can be extrapolated to aReynolds number lower than 10 with a good approximation.

Stokes’s law defines the force applied on an object in a uniform flow of fluiddefined by a dynamic viscosity μ3 and a velocity V :

Fhydro = −kμ3 V (2.40)

where k is a function of the geometry. In case of a sphere with a radius R, k isdefined by

k = 6πR

Table 2.1 gives the values of dynamic viscosity μ of both water and air. So, thehydrodynamic force proportional to the dynamic viscosity highly increases in asubmerged medium.

2.2.3 Theoretical Comparison Between Air and Liquid

2.2.3.1 Surface ForcesWhen two media are in contact, the surface energy W12 is equal to

W12 � 2√

γ1γ2 (2.41)

with γi the surface energy of the body i.From the previous formulas the energy W132 required to separate media 1 and

2 immersed in medium 3 is given by

W132 = W12 + W33 − W13 − W23 = γ13 + γ23 − γ12

For example, in case of a silicon–silicon contact (γsilicon = 1400 mJ · m−1),surface energy in water and in air are:

W12 = 2800 mJ m−1 W132 = 1670 mJ m−1 (2.42)

TABLE 2.1. Density and Dynamic Viscosity of Waterand Air, T = 20◦C

Properties Water Air

ρ (kg m−3) 103 1, 2μ (kg m−1 s−1) 10−3 18.5 × 10−6


TABLE 2.2. Values of Hamaker Constant for SomeMaterials A/10−20J

Materials Vaccum Water

Gold 40 30Silver 50 40Al2O3 16.75 4.44Copper 40 30

In this example, the pull-off force is less in water compared to air. Usually,solid-state surface energy is around 1000 mJ m−1, and this example gives a goodoverview of the reduction of pull-off forces in a liquid medium.

2.2.3.2 Van der Waals ForcesFor contact of two dissimilar materials in the presence of a third media, A132

may be estimated by

A132 = A12 + A33 − A13 − A23

Using combination formulas, it follows that

A132 ≈ (√

A11 −√

A33)(√

A22 −√

A33) (2.43)

Table 2.2 gives the values of the Hamaker constant for some materials invacuum and in water.

2.2.3.3 Electrostatic ForcesThe force applied by an electrostatic surface (σ surface charge density) on anelectrically charged particle (q) is given by

F = qσ

2ε0ε(2.44)

whereε = relative dielectric constant of the medium

ε0 = dielectric constant of the vaccum

The relative water dielectric constant (ε = 80.4) is greater than the relativeair dielectric constant (ε = 1). So, in the same configuration of electrical charges(q, σ ) the electrostatic force is less in water.

Moreover electrostatic perturbations observed in micromanipulation are causedby triboelectrification. During a microassembly task, friction between manipu-lated objects induces electric charges on the object’s surface. The charge densityis a function of the triboelectrification and conductivity of the medium. Effec-tively, a higher electric conductivity medium is able to discharge the object


surface. Water, especially ionic water, has better electric conductivity than air.Consequently, the charge density in water is less. The electrostatic force directlyproportional to charge density is therefore less.

In conclusion, electrostatic perturbation forces are much lower in water com-pared to air.

2.2.4 Impact of Hydrodynamic Forces on Microobject Behavior

We present in this section the impact of the hydrodynamic forces on the behaviorof microobjects and especially on the reduction of the objects’ jumps.

Because inertial effects are negligible in the microworld, microobject accelera-tions are usually very high. In this way, microobject velocity is able to increase ina very short time. Consequently, microobjects can reach high velocity, and objecttrajectory could be difficult to control especially because of the visual feedback.So, in most of cases, velocity limitation in the microworld is not induced byinertial physical limitation but by hydrodynamic physical limitation. From thispoint of view, a liquid medium is able to reduce maximal microobject velocity.

As a synthetic example we prove in this part that the hydrodynamic forcesallow to reduce significantly the jump of microobjects.

To study the diminution of object jump, we have considered the trajectory ofa microball (diameter d and density ρ0), which has an initial velocity V (0):

V (0) = Vx(0) x + Vz(0) z (2.45)

The hydrodynamic force and weight are applied to the object (see in Fig. 2.4).We compare the object trajectories in air and in water. Table 2.1 recalls thedensity ρ and dynamic viscosity μ values of both water and air

In the case of an object placed in a uniform liquid flow, Stokes’s law directlygives the hydrodynamic force applied on the object. Stokes’s law is valid whenthe Reynolds flow number is lower than 1 and can be extrapolated to a Reynoldsnumber lower than 10 with a good approximation.

Ball velocity: V(0)

Ball diameter: d

Sufficient distance to neglect the action of the substrate

Ball

G

z

x

y Substrate

Figure 2.4. Example of hydrodynamic effects on microobject behavior: initial configura-tion.


We consider that Stokes’s law (2.40) is valid. The object trajectory obtainedwith the ball dynamic equilibrium equation verifies⎧⎪⎨

⎪⎩x(t) = Vx(0)τ (1 − e−t/τ )

z(t) = Vz(0)τ

[− t

τs

+(

1 + τ

τs

)(1 − e−t/τ )

](2.46)

with ⎧⎪⎪⎨⎪⎪⎩

τs = ρ0

(ρ0 − ρ)

Vz(0)

g

τ = ρ0d2

18μ

(2.47)

Parameter τ represents the time constant associated with the hydrodynamicforce. Parameter τs represents the time constant associated with gravity (sedi-mentation).

In water, τ is much smaller than τs :

τ � τs (2.48)

In other words, the impact of the hydrodynamic force on the trajectory is fasterthan the impact of gravity (sedimentation). So the trajectory is divided into twosteps:

t ∼ τ

⎧⎨⎩

x(t) = Vx(0)τ (1 − e−t/τ )

z(t) = Vz(0)τ (1 − e−t/τ ) = Vz(0)

Vx(0)x(t)

(2.49)

t ∼ τs

⎧⎨⎩

x(t) = Vx(0)τ

z(t) = Vz(0)τ

(1 − t

τs

)(2.50)

We obtain a “triangle” trajectory:

• First stage: The trajectory is linear with its direction being the initial velocity V (0).

• Second stage or sedimentation stage: The trajectory is linear with its direc-tion being gravity (−z).

Also the final object position L is

L = Vx(0)τ from (2.50)

= ρ0Vx(0)d2

18μfrom (2.47) (2.51)


We consider that this position is reached at t = T = 5τ :

T = 5ρ0d2

18μfrom (2.47) (2.52)

We have chosen to present the following example:⎧⎪⎨⎪⎩

ρ0 = 2000 kg m−3

d = 50 μm

Vx(0) = Vz(0) = 35 mm s−1

(2.53)

The final position calculated by Eqs. (2.51) and (2.52) is

In water L = 10 μm at T = 1.4 ms (2.54)

In air L = 530 μm at T = 75 ms (2.55)

Object trajectories are described in Figure 2.5. In the case of water[Fig. 2.5(a)], simulated trajectory (2.46) and “triangle trajectory” 2.49 and 2.50are similar. In the case of air [Fig. 2.5(b)], hypothesis 2.48 is not true and thetrajectory is different from the “triangle trajectory.” However, Eqs. 2.51 and2.52 remain valid.

This example illustrates microobject behavioral differences between the liquidmedium and the dry medium. The distance L (defined in Eq. mk) is inverselyproportional to the dynamic viscosity μ, and is 50 times longer in air than inwater. The displacement of the microobject in air is around 10 times its diam-eter contrary to water, where the displacement is about one fifth of the object’sdiameter. This example illustrates the impact of the hydrodynamic forces on thereduction of microobject loss.

2.2.4.1 ConclusionTo explain the experimental differences between dry micromanipulation andmicromanipulation in liquid, we analyze in this section the theoretical impactof the liquid on surface forces. We will focus this study on water. The currentsurface forces considered in the microworld are capillary, van der Waals, pull-off,and electrostatic forces. Although the main forces that induce adhesion effectsare pull-off and electrostatic forces, we have chosen to present in this section theinfluence of water only on these two forces.4 We also present the impact of thehydrodynamic forces on the microobject’s behavior. We propose to separate theseforces by making the distinction between whether there is contact or not. Whenthere is no physical contact between two solids, the forces in action are calleddistance or surface forces (according to the scientific literature in this domain [9,15, 45], the latter are electrostatic, van der Waals, and capillary forces). When

4We could also prove that van der Waals forces and capillary forces are, respectively, reduced [62]and decreased in a liquid medium.


0 2 4 6 8 10× 10−6

0

1

2

3

4

5

6

7

8

9

10× 10−6

x (m)

z (m

)

L = 10 µm

(a)

−1 0 1 2 3 4 5 6× 10−4

0

1

2

3

4

5

× 10−4

x (m)

z (

m)

L = 530 µm

(b)

Figure 2.5. Comparison between ball trajectories in water and in air: (a) ball trajectory inwater similar to the triangle trajectory and (b) ball trajectory in air compared to a triangletrajectory.


both solids are in contact with one another, there are deformation and adhesionforces through the surfaces in contact. In this case, we consider contact forcesand adhesion or pull-off forces. Electrostatic or capillary effects can be added,but van der Waals forces are not considered in this case because they are alreadyincluded in the pull-off term.

2.3 MICROSCOPIC ANALYSIS

2.3.1 AFM-Based Measurements

The goal of this section is to compare theoretical noncontact and contact surfaceforces at the microscale with corresponding values measured with an atomicforce microscope (AFM). In order to develop new and adequate systems forefficient and reliable manipulation systems for objects at the microscale, it isindeed really necessary to be able to correctly estimate the forces at play througha set of realistic equations describing these forces.

2.3.1.1 DescriptionA view of the device is given in Figure 2.6(a). This system is based on anatomic force microscope. The microlever of the AFM can be moved in threeperpendicular directions XYZ5 by a piezotube (with X, Y , and Z strokes of,respectively, 45, 45, and 4 μm). Three linear micropositioning stages are also usedfor the studied sample motions on longer strokes in XYZ directions (15 × 15 × 15mm3, with a repeatability of 0.1 μm). All these motions can be controlled inautomatic mode or in manual mode, notably by using a force-feedback joystick.This joystick applies in real time to the operator, with the bending and torsioneffects measured on the microlever. These strains are measured by a photodiode(which delivers a corresponding voltage Vm).

Finally, two microscopes with charge-caupled device (CCD) cameras givevisual information in real time on, respectively, vertical and lateral views [seeFig. 2.6)(a)]: One is fixed under the glass sample support and the other is placedperpendicularly. Then the first one gives the position of the microlever tip in thesample plan. The second one is used to estimate the vertical distance betweenthe tip and the sample plan.

In this book, the microlever used is a silicon one that is 350 μm in length,35 μm in width, and 2 μm in thickness. Its tip [see Fig. 2.6(b)] has a curvatureradius of less than 10 nm and a height of 15 μm.

2.3.1.2 Measurement MethodOur setup is used here for the production of experimental force curves based onthe real-time measurement of the AFM microlever bending (Fig. 2.7). A forcecurve is a quasi-static trajectory that corresponds to an ”approach-and-retract”cycle between the microlever and the sample (in the Z direction). Depending on

5The Z axis designating the vertical direction.

MICROSCOPIC ANALYSIS 75

AFM

Sample

(a)

(b)

Microscope 2

Microscope 1

Figure 2.6. Description of atomic force microscope measurement system: View of (a)the AFM and (b) the tip.

the required stroke, these motions are actuated by the piezotube as well as thesample vertical axis.

First, the sample is placed in the AFM system and the operator can start theapplication: The approach-and-retract cycle is then executed automatically. Theacquired data are the microlever bending, that is, the voltage measured from thephotodiode, according to the relative vertical motion between the microlever and


0.08

0.06

0.04

0.02

0

Nor

mal

forc

e (μ

Ν)

Lever position (μm)

Approach

Deflection curve

Retract

7

12

5

4

3 6−0.02

−0.04

−0.06

−0.08

−0.1

−0.12−10.5 −10 −9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6

Figure 2.7. Example of force curve.

the sample. Then an adequate processing is done to extract the bending forcesF from the measured voltage Vm. A specific identification procedure is used toestimate the coefficient C, which linearly relates Vm to the microlever bendingδ and the microlever stiffness k. The stiffness of the microlever used for theexperiments is k = 0.03 N/m. Then F is obtained by

F = kδ = kCVm (2.56)

An example of such a curve is shown on Figure 2.7

2.3.2 Experiments on Adhesion Forces

2.3.2.1 Van der Waals Forces and Pull-Off ForcesFirst, the AFM measurement device is used to study van der Waals forces(attraction in approach phase of the AFM tip) and pull-off force (breaking loadduring the withdrawal of the AFM tip). These experiments are carried out withpolystyrene and glass substrates.

From these curves [see figs. 2.8(a) and 2.8(b)], an approached value of thepull-off forces for these two interactions is deducted. They are estimated at

Psilicon–polystyrene = 26.23 nN (2.57)

Psilicon–glass = 34.70 nN (2.58)


25

20

15

10

5

0

0 0.5Displacement (μm)

(a)

1 1.5 2 2.5 3

Effo

rt (

nN)

1

2I

0−40

−30

−20

−10

0

10

20

30

0.5 1 1.5 2Displacement (μm)

For

ce (

nN)

2.5 3

(b)

Figure 2.8. Force curves: interactions between AFM tip and (a) a polystyrene substrateand (b) a glass substrate.

The following data are used to determine the theoretical values of pull-offforces:

• Silicon γ = 1400 mJ/m2, A = 25.6 × 10−20 J, ν = 0.17, and E = 140 MPa.• Polystyrene γ = 36 mJ/m2, A = 7.9 × 10−20 J, ν = 0.35, and

E = 3200 MPa.• Glass γ = 170 mJ/m2, A = 6.5 × 10−20 J, ν = 0.25, and E = 69,000 MPa.

From Eq. 1.105, pull-off forces can be estimated:


• Silicon–polystyrene interface λ = 0.33, andP = 28.21 nN.• Silicon–glass interface λ = 0.54, andP = 39.43 nN.

These values fit very closely to the measurements. Hence, theoretical estima-tion of pull-off forces can generally be trusted when no direct measurement ispossible.

On these force curves, one can observe the attraction phenomena in theapproach phase. Though some works suggested different origins to the behavior,a traditional approximation is to consider only van der Waals interaction [43, 61].These forces are thus estimated at:

• Silicon–polystyrene interface F = 5.14 nN.• Silicon–glass interface F = 4.03 nN.

With the assumption of the contact distance D0 = 0.165 nm, the values ofHamaker constants of the interfaces can be deduced from measurements:

• Silicon–polystyrene interface A = 8.43 × 10−20J.• Silicon–glass interface A = 6.58 × 10−20J.

These measured values can be compared with the values obtained from theequation A12 ≈ √

A11A22:

Asilicon–polystyrene = 13.87 × 10−20 J (2.59)

Asilicon–glass = 13.01 × 10−20 J (2.60)

The errors are more significant because it is difficult to determine the variousphenomena [53]. Nevertheless, this estimation seems to give a realistic value inorder to estimate these forces via the Hamaker constants. A significant remark isthat the range of van der Waals forces are of the order of 100 nm for all experi-ments carried out. Thus, this force seems to be relatively negligible compared toforces for objects of microscopic size.

A last experiment studies an interaction with a glass substrate in an aqueousenvironment in order to see the influence of the environment (Fig. 2.9).

The pull-off force is thus estimated at

P = 5.52 nN (2.61)

The theoretically calculated pull-off is then

P = 8.06 nN (2.62)

The estimation is still rather precise. Note also that the van der Waals force isalmost not perceptible any more by our system. Its influence thus seems negligiblein an aqueous environment.


Displacement (μm)

For

ce (

nN)

−60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

−4

−2

0

2

4

6

8

Figure 2.9. Force curve for an interaction between the cantilever and a glass substrate ina aqueous environment.

2.3.2.2 Capillary ForcesFor micromanipulation, the capillary forces can represent a key parameter. Twointeractions are studied with our system by placing a water droplet on polystyreneand glass substrates (Fig. 2.10). As the height of the droplet exceeds the maximumstroke of the AFM probe actuator, only the table motion is possible. Note thatin this case force curves are reversed in terms of displacement and also that thismotion reduces precision.

The capillary force is dependent on the reverse of the separation distance,contrary to the van der Waals and electrostatic forces. This force is present onlywhen the AFM probe actually touches the water droplet. This force can thus beestimated using the following contact angles [13]

θglass = 37◦θpolystyrene = 67◦ (2.63)

The measured forces are about

Fsilicon–glass = 71.98 nN (2.64)

Fsilicon–polystyrene = 37.82 nN (2.65)

The theoretical forces can thus be calculated from the equationFcap = 4πRγl cos θ/(1 + D/d) [28]:

Fsilicon–glass = 73.26 nN (2.66)

Fsilicon–polystyrene = 35.86 nN (2.67)

Theoretical models fit the capillary interaction quite well. The followingremarks can be proposed for the sensivity of the capillary forces:


−45

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

−90−40 −35 −30 −25 −20 −15 −10 −5 50

For

ce (

nN)

Displacement (μm)

(a)

20

10

0

−10

−20

−30

−40−16 −14 −12 −10 −8 −6 −4 −2 0

Displacement (μm)

For

ce (

nN)

(b)

2 4

Figure 2.10. Force curves for interactions between the AFM tip and a water droplet: (a)on a polystyrene substrate and (b) on a glass substrate.

• When they exist, the capillary forces are most significant (their module ishigher than van der Waals module).

• Their range is almost null.• They always exist in a laboratory environment via, for example, a layer of

oxidation of around of 10 nm on metals [60]. This layer seems negligiblecompared to the size of the considered object. They are commonly in theexpression of pull-off forces.

Moreover, the pull-off force is overbalanced by a viscoelastic force due to thepresence of water.


10

5

0

−5

−10

−15

−20

−25

−30

−35

−40−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0

Displacement (μm)

(a)

For

ce (

nN)

10

5

0

−5

−10

−15

−20

−25

−30

−35

−40−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1

Displacement (μm)

(b)

For

ce (

nN)

Figure 2.11. Force curves for interactions with (a) gold substrate and (b) grounded sub-strate.

2.3.2.3 Electrostatic ForcesThis last part studies the electrostatic forces in case of contact with conductors andinsulators. The AFM tip is made of silicon and is grounded. The first experimentdescribes a contact with a gold substrate. The electrostatic forces appear at a verysignificant separation distance compared to the other forces (10 μm).

To avoid this force, the substrate can be grounded as in Figure 2.11(b). In thesame way, the van der Waals forces are not measurable (its range is about theresolution of the microtranslator).

The second study is on an insulator of polystyrene substrate. The results areshown in Figure 2.12(a).


10

5

0

−5

−10

−15

−20−25 −20 −15 −10 −5 0

Displacement (μm)

(a)

For

ce (

nN)

10

5

0

−5

−10

−15

−20

−25

−30

−35

−40

−45−14 −12 −10 −8 −6 −4 −2 0 2

Displacement (μm)

(b)

For

ce (

nN)

Figure 2.12. Force curves for interactions with polystyrene substrate: (a) interactions witha polystyrene substrate and (b) interactions with a polystyrene substrate after cleaning thesubstrate with distilled water.

In the same way, to avoid this force, the substrate is cleaned with distilledwater. The curve obtained is then represented in Figure 2.12(b).

Electrostatic forces are efficient in the long range, starting at 10 μm. As theelectrostatic load of the micropart is very badly known, and gives the highestmodules of the distance forces, the most suitable approach for a micromanipula-tion application is to avoid this phenomenon by grounding for conductor or bythe use of distilled water for the insulator. Note that in case of a manipulationinvolving only conductors, including the gripper, the use of electrostatic forcecan be an interesting solution.


15

10

5

0

−5

−10

−15

−20

−25

−300 0.5 1

Displacement (μm)

(a)

For

ce (

nN)

1.5 2 2.5 3

20

10

0

−10

−20

−30

−40

−50−25 −20 −15 −10 −5 0

Displacement (μm)

(b)

For

ce (

nN)

Figure 2.13. Force curves for electrostatic forces: (a) and (b) interaction with a glasssubstrate.

2.3.3 Various Phenomena

Two complementary experiments are finally proposed. The first describes aninteraction with a glass substrate. On this curve (see Fig. 2.13), it is possibleto observe at the same time the appearance of the electrostatic force (repulsiveforce with glass) and close to the contact, the appearance of the attractive van derWaals force. The amplitudes of these forces are comparable. Thus, the capillaryforces, if they exist, are the most significant compared to electrostatic and vander Waals forces.

The second experiment studies the approach of the AFM cantilever with acopper substrate initially charged with a 2-V voltage. The approach curve of the


AFM cantilever is then drastically modified (see Fig. 2.13). Phases of attrac-tion/release appear due to phases of discharge of the AFM tip. Moreover, tipeffects can be observed, making difficult any identification. The only means ofcarrying out a discriminating analysis is to use a tipless cantilever. In the sameway, this phenomenon disappears as soon as the substrate is grounded.

A lot of measurements are taken in this section. They highlight significantand often ignored phenomena as the influence of the force of pull-off, the weakrange of the van der Waals forces, the influence of the capillary forces, and thelong range of the electrostatic forces.

2.4 SURFACE ROUGHNESS

This section is a brief introduction to some of the concepts linked to surfaceroughness (measure, modeling, manufacturing). Additional information on howto include surface roughness in surface forces studies can be found in SausseLhernauld [50]. Indeed, it turns out that roughness cannot be avoided (effectson electrostatic forces have been proofed for as small a roughness as a fewnanometers [51]). Therefore, it is of the utmost importance for engineers to beable to measure and model it. The following introduction will be restricted to 2Dgeometries. Details for 3D geometries can be found in tribology books such asBhushan [6].

2.4.1 Surface Topography Measurements

Experimental techniques used for surface topography measurements can bedivided into two broad categories: contact types and noncontact types (seeTable 2.3) [6]. Different techniques are briefly reviewed in this section:

• Mechanical stylus method uses an instrument that amplifies and recordsthe vertical motions of a stylus displaced at a constant speed above thesurface to be measured. As the stylus rides over the sample, detecting surfacedeviations by a transducer, it produces an analog signal corresponding tothe vertical stylus movement that is amplified, conditioned, and digitized.

TABLE 2.3. Comparison of Roughness Measurement Methods [6]

Spatial VerticalMethod Resolution Resolution Limitations

Stylus 15–100 nm 0.1–1 nm Contact maydamage surface.

Optical 0.5 μm–1 mm 0.02–25 nmSTM 0.2 nm 0.02 nm Conductive surfacesAFM 0.2–1 nm 0.02 nmSEM 5 nm 10–50 nm Conductive surfaces

SURFACE ROUGHNESS 85

• Optical microscopy uses light waves reflected on the surface. The angleof reflection is equal to the angle of incidence in the case of perfectlysmooth surfaces. As roughness increases, the intensity of the specular beamdecreases while the diffracted radiation increases in intensity and becomesmore diffuse. Optical methods may be divided into geometrical (taper sec-tioning and light sectioning) and physical methods (specular and diffusereflexion, speckle pattern, and optical interference).

• Scanning tunneling microscopy (STM) is based on the principle that if apotential difference is applied to two metals separated by a thin insulatingfilm, a current will flow because of the ability of electrons to penetrate apotential barrier. In STM a sharp tip (one electrode of the tunnel junction) isbrought close enough to the surface to be investigated (second electrode) sothat, at a convenient operating voltage, the tunneling current will vary andwill be measured. STM requires the surface to be measured to be conductive.

• Atomic force microscopy (AFM) is capable of investigating surfaces ofboth conductors and insulators on an atomic scale. AFM measures ultra-small forces (less than 1 nN) present between the AFM tip and a surfaceto be investigated. These forces are measured by measuring the motion ofa very flexible cantilever beam.

• Scanning electron microscopy (SEM) uses a beam of highly energetic elec-trons to examine objects on a very fine scale. It functions exactly as itsoptical counterparts except that SEM uses a focused beam of electronsinstead of light to image the specimen and gain information about its struc-ture and composition.

Other methods to measure surface roughness include fluid methods and elec-trical methods that are mainly used for continuous inspection procedures (qualitycontrol) because they are very fast and function without contact with the surface.

A typical encountered problem in roughness measuring is the influence of thecurvature radius of the scanning probe (see Figs. 2.14 and 2.15. The probe filterscurvatures: Only curvatures smaller than that of the probe can be measured (alarge tip cannot scan into small roughness valleys).

2.4.2 Statistical Parameters

Two main approaches can be followed to describe roughness, relying on statistical(described in this section) or fractal (described in Section 2.4.4) descriptions.

The statistical approach relies on the computing of several statistical amplitudeand spatial parameters in order to fit a measured profile z(x). For a profile scannedalong a sampling length L, the parameter Ra (center-line average) is given by

Ra = 1

L

L∫0

|z(x) − z| dx (2.68)


0 0.2 0.4 0.6 0.8 1 1.2

× 10−6

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

× 10−7

x (m)

z (m

)

Figure 2.14. Influence of the curvature radius of the scanning probe: actual profile (solidline), measured profile (dashed line), circles indicate three different locations of the probe(curvature radii equal to 300 and 50 nm).

Figure 2.15. Influence of the curvature radius of the scanning probe: actual profile (solidline), measured profile (dashed line).


where z is the average height of the profile. The standard deviation of the distri-bution z(x) is the base of Rq :

Rq =√

1

L

∫ L

0(z(x) − z)2 dx (2.69)

Both parameters can be used to define a Gaussian model of the surface, butunfortnuately, not all manufacturing processes lead to such a profile [40]. In thiscase, additional indicators can be used such as those built on third- (skewness)and fourth- (kurtosis) order moments [7].

Norms also mention the maximum peak-to-peak distance Rt , which is unfor-tunately not robust (it can be very sensitive to the presence of dust):

Rt = max(z) − min(z) (2.70)

We see from these definitions the sensitivity of these amplitude parameters tothe sampling length L [30]: These parameters are not intrinsically related to thesurface [63].

One way to supplement amplitude information is to provide some index ofcrest spacing or wavelength on the surface [6]. Such parameters include theaverage wavelength and the root-mean-square (rms) wavelength defined by

λa = 2πRa

δa

(2.71)

λq = 2πRq

δq

(2.72)

where δa and δq are the amplitudes of the individual wavelength. Other parame-ters may include the peak density or the mean spacing between peaks. The rmsvalues are, however, still scale dependent.

Beside these parameters, additional functions can be used: They are alsoknown as surface texture descriptors and referred to as spatial function [5]:

1. The autocorrelation function is a way of representing spatial variation. Itis the product of two measurements taken on a profile at a distance τ apart.

2. The structure function is defined by

S(τ) = limL→∞

1

L

∫ L

0[z(x) − z(x + τ)]2 dx (2.73)

3. The power spectrum (or power spectral density) describes how the powerof a signal or time series is distributed with frequency. It is deduced fromthe square of the modulus of the Fourier transform of the signal:

P(ω) = limL→∞

1

L

[∫ L

0z(x) exp(−iωx) dx

]2

(2.74)


Inversely, the autocorrelation function is the inverse Fourier transform ofthe power spectrum:

S(τ) =∫ ∞

−∞{[1 − exp(iωτ)]P(ω)} dω (2.75)

An alternative is to consider a representation of the surface, which is independenton the considered scale, which is precisely one of the features of fractal modelingknown as self-similarity. This will be the topic of Section 2.4.4.

2.4.3 Models of Surface Roughness

When studying the adhesion phenomenon, surface roughness is a very importantfactor. Indeed it is well known that the existence of nanoscale roughness dra-matically reduces adhesion between surfaces and that this is due to a decreasein the real area of contact and an increase in the distance between bulk surfaces[33, 57]. For these reasons many attempts have been made to model surfaceroughness that will be summarized in the following. Please note that much morework has been done in the field of influence of surface roughness on van derWaals forces since roughness has a great influence on these forces, whereas theelectrostatic forces are often believed not to change much with asperities [9].

The simplest case [2] only considers the roughness peak and assumes theroughness profile to be equivalent to a smooth profile located at a separationdistance d+R/2 where R is the height of the highest peak and d is the distancebetween the plane and the highest peak. This model is, however, not good sinceit does not take the density of protrusions into account.

Sphere (hemispherical model) and cone models are the most encountered inthe literature:

• Vogeli et al. [61] model the roughness profile using several half-sphereswith a diameter R.

• Herman et al. [24] determine the effects of spherical and conical asperitieson van der Waals and double-layer interactions between two parallel flatplates.

• Suresh et al. [25] model roughness as hemispherical asperities characterizedby the average asperity height and the density of asperities on the surface forcalculating both electrostatic and van der Waals interaction energies betweena rough particle and a flat plate.

Discretized profiles have also been used as a roughness representation:

• In Lambert [34] the Abbott diagram is considered and related to the surface,the roughness profile is discretized into M cites, and his equation is appliedto the M -discretized elements (see Fig. 2.16).


• In Shulepov and Frens [52] the roughness of the particle is characterizedas a uniform pattern of hills, each hill consisting of steps and plateaus witha characteristic width and height to evaluate the total interaction energybetween two rough spheres.

Sinusoidal functions are also a way of representing roughness, knowing theamplitude and the period (or as a sum of sine function):

• In Danuser et al. [12] a sinusoidal surface is modeled, where the density nof protrusions is related to the wavelength λ by n = 1/λ2 and the radius ofthe spheres is r = λ2/(4π2h) (see Fig. 2.16).

• In Kostoglou and Karabelas [32] a cosine function is used that is an idealizedperiodic surface characterized by shape, height, and wavelength to calculatethe electrostatic repulsive energy between two rough colloidal particles.

The limitation of all the previous models is that the roughness is representedas a very idealized shape. Such approaches provide nonetheless qualitative infor-mation about the effects of surface roughness. Unfortunately, little has been saidabout how well those geometries correlate with known surface roughness pro-files, especially at the nanoscale. Moreover, Rabinovich et al. [47] state thatthe application of a Gaussian distribution to model asperities on a surface mayproduce errors because large asperities significantly affect the interaction eventhough their number is low. Another modeling strategy is therefore presented innext section.

2.4.4 Fractal Parameters

2.4.4.1 Introduction to FractalsBecause surface roughness can have such a large effect on the adhesion force,errors introduced by modeling surface roughness with elementary protuberanceswill cause adhesion force predictions to be inaccurate in many cases [19]. Moreparticularly, a roughness description based only on the Ra parameter puts asidespatial consideration. An alternative is to proceed to a Fourier analysis of thesampled profile in order to extract the main components of the roughness signal,leading to the following mathematical model of the surface:

z(x) =nf∑i=1

A(i) cos

(2π

x

λi

)(2.76)

where the λi is the spatial wavelength of the ith component with amplitude Ai .This approach can be very powerful but still requires 2nf parameters to recon-struct the geometry. Nevertheless, some surfaces exhibit a self-repeating nature ofsurface roughness at different scales: They are known as fractal surfaces. The termfractal comes from the Latin fractus , meaning irregular or fragmented. Fractalsare irregular objects possessing similar geometrical characteristics at all scales.This characteristic is called self-similarity . Observing the coastline of Britain,


RaRa

Sk

ts

S

L

D

Sk

S

rkRS

1

1

L

D

S

Z

h

rl

R

dd + R/2

(a)

d

R

(b)

(d)

(c)

Figure 2.16. Discretized profiles from (a) planar [2], (b) hemisphere [61], (c) discretized[34], and (d) sinus [12].

Figure 2.17. Illustration of the self-similarity property [63].

Mandelbrot [38] showed, for example, that the more the coastline is magnified,the more features and details are observed. This illustrates the self-similaritynotion (Fig. 2.17).

Geometries of fractal surfaces are also continuous and nondifferentiable. Sincethe profile of rough surfaces z(x) (typically obtained from stylus measurements)is assumed to be continuous even at the smallest scales and ever-finer levelsof detail appear under repeated magnification, the tangent at any point cannotbe defined. The profile has thus the mathematical property of being continuouseverywhere but nondifferentiable at all points. Surface profiles are also knownto have self-affinty in roughness structure. The Weierstrass–Mandelbrot function


satisfies the properties of continuity, nondifferentiability, and self-similarity andcan therefore be used to simulate such profiles, with only a small number ofparameters.

2.4.4.2 Fractal Representation of RoughnessIn fractal characterization, the continuity, nondifferentiability, and self-affinityof a two-dimensional surface profile height may be represented by the function[8, 18, 23, 31, 37, 44, 63, 64, 68]

z(x) = GD−1∞∑

n=n1

cos(2πγ nx + φ)

γ (2−D)n(2.77)

where D is the fractal dimension (1 < D < 2), G is the fractal roughness param-eter or scaling parameter, φ is random phase, and L is the fractal sample length.

Constant γ is chosen to be 1.5 [23, 31, 37] for phase randomization and highspectral density (in order for the phases of the different modes not to coincideat any given x position, the value of γ must be chosen to be a noninteger).Constant γ determines the density of the frequency spectrum with smaller γ

values yielding larger numbers of frequency components.Equation 2.77 represents the surface profile by a series of cosine functions

(Fig. 2.18) with geometrically increasing frequencies starting from the lowestfrequency wl = 1/L. Factor n1 is thus deduced from the length of the sample L:γ n1 = 1/L.

Figure 2.18. Construction of a fractal surface [56].


2.4.4.3 Description of Fractal ParametersFractal parameters are D and G. Parameter D can vary from 1 to 2 in two dimen-sions, whereas G is not limited. As D becomes larger, the number of asperitiesincreases and their height decreases; therefore, it governs the contribution of low-and high-frequency components to the surface (see Figs. 2.19 and 2.24). Theamplitude or scale parameter is G (also called fractal roughness). As it increases,the peaks and valleys are amplified. Thus, as the magnitudes of D and G increase,a rougher and more disordered surface topography is produced. An increase ofD stretches the profile in the lateral dimension and therefore changes the spatialfrequency. On the other hand an increase of G stretches the profile along thevertical dimension. So D controls the relative amplitude of roughness at differ-ent length scale, whereas G controls the amplitude of roughness over all lengthscales. The Hausdorff or fractal dimension, D + 1, of rough surfaces in threedimensional simulations is a fraction between 2 and 3 [36]. The specific integervalues of 0, 1, 2, and 3 correspond to smooth objects, respectively, point, line,surface, and sphere (or any three-dimensional object), whereas the nonintegervalues correspond to wiggly and complex objects with self-similar behavior.

2.4.4.4 Advantages/LimitationsAdvantages. The advantage of fractals is to offer a set of parameters that areinvariant with respect to scale. More specifically, once scale-invariant parame-ters have been determined, the roughness can be predicted at all length scaleswith only two parameters (D and G). Characterizing a multiscale surface usingtraditional statistical parameters would render different results depending on thescale at which the measurements were taken. Of course, all surfaces are notfractal, and obviously the fractal description can only be applied to surfaces thatshow fractal behavior. According to Morraw [41], most engineering surfaces havebeen proved to exhibit roughness on different scales. For Komvopoulos [30], thetopography of many engineering surfaces may be represented by fractals becausesimilar features can be observed at different magnification of the same surface.In Ling [35], conclusions are that fractal geometry forms an attractive adjunct toEuclidean geometry in the modeling of engineering surfaces. For Majumdar andBhushan [36], it is necessary to characterize rough surfaces by intrinsic parame-ters that are independent of all scales of roughness (see Fig. 2.20). This suggeststhat the fractal dimension, which is invariant with length scales and is closelylinked to the concept of geometric self-similarity, is an intrinsic property andshould therefore be used for surface characterization.

For Zhou et al. [67] manufactured surfaces produced by electrical discharge,water-jet cutting, and ion-nitriding coating can be characterized by fractal geome-try. It is moreover important to introduce fractal-based techniques to study surfaceengineering and tribology and to apply the techniques to engineering applicationssuch as contacts, wear processes, and friction. It has also been shown that sur-faces formed by natural and random courses such as fracture surface of a solid, adeposition surface of a material, or a solidification surface of a liquid have fractalstructures [23]. All the previously cited works tend to validate the theory that


fractal geometry gives an accurate description of the profile geometry for manyengineered surfaces. This point of view is not, however, shared by all authorsand is discussed in the next section.

Limitations. The scale independence of the fractal parameters D and G wasdoubted by Ganti and Bhushan [18]. In a series of measurements of fractalparameters, measures have shown to depend on the method used for surface scan-ning (i.e., AFM or nontcontact optical profiler) and on the scan size. Majumdarand Bhushan [36] even discovered that some rough surfaces are bifractal, thatis, the surface exhibits different D and G values within different scale ranges.These studies bring the conclusion that fractal parameters are relatively scaleindependent but not absolutely scale independent in all scale ranges [23].

Gelb et al. [19] used fractals to model rough copper surface and polytetraflu-oroethylene (PTFE) surfaces. In both cases, the fractal dimension decreased withdecreasing the scan size, indicating that fractals should not be used to describethese surfaces.

Because fractals are scale invariant, they are hardly suitable for characteriz-ing features that are not. Only if the surface feature is scale invariant can it beexpected to reflect the dimension D. Whitehouse [63] does not agree with the factthat engineered surfaces exhibit fractal properties. The author states that fractalanalysis can be used to characterize natural phenomena such as growth mecha-nisms (e.g., bifurcation or fracture mechanics), but most manufacturing effects arenot growth phenomena. The conclusion is that fractals tend to ignore manufacturerather than to clarify it. Whitehouse thinks random process parameters are moresuitable to describe manufacturing mechanisms. In fact, whitehouse [63] evenstates that trying to allot fractal parameters to surfaces is questionable. Indeed,he thinks using scale-invariant parameters as a replacement for traditional param-eters (statistical parameters) is philosophically wrong. He warns that fractals inmanufacture are not necessarily the best means of characterization. Because man-ufacturing characteristics are severely scale limited, surface parameters that aredeliberately scale dependent should be used.

Real engineering surfaces might be multifractal or be fractal on certain lengthscales and nonfractal on others. It is difficult to treat the fractal parameters D andG as absolutely scale independent on all scale ranges [23]. However, as theseparameters are scale independent on certain length scales (i.e., relatively scaleindependent), we might still assume that the fractal model is applicable.

Concerning the effect of the sampling interval on the fractal parameters, exper-imental results from He and Zhu [23] show that changes in sampling spacing(lateral resolution) did not influence the scale independence of fractal parametersand that if the sampling length is long enough, it will have no influence on thefractal parameter.

2.4.5 Extracting the Fractal Character of Surfaces

Fractals have many dimensions such as Hausdorff dimension, compass dimen-sion, box dimension, mass dimension, and area perimeter dimension, and there are


0 0.2 0.4 0.6 0.8 1 1.2−8

−6

−4

−2

0

2

4

6

8 × 10−3 × 10−8

× 10−3

D = 1.2G = 1.5 × 10−12

0 0.2 0.4 0.6 0.8 1 1.2−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2−5−4−3−2−101234

× 10−6

0 0.2 0.4 0.6 0.8 1 1.2−5−4−3−2−101234

D = 1.7G = 1.5 × 10−12

G = 3 × 10−12

D = 1.5G = 3 × 10−6

D = 1.5

Figure 2.19. Influence of parameters D and G on a 1-μm-long profile; axis are in meters.

several methods for computing each of these dimensions [67]. In what follows,we detail two methods—power spectrum and structure function methods—inorder to determine the parameters D and G to be used in Eq. [2.77].

2.4.5.1 Power Spectrum MethodThe variation of roughness amplitude with spatial frequency ω of theWeierstrass–Mandelbrot function can be represented by the averaged powerspectral density function P(ω). It is calculated using the real surface profile [46]:

P(ω) = 2π x

N|F(ω)|2 (2.78)

where x is the spacing between two points, ω is the spatial frequency, N is thenumber of points sampled, and F(ω) is the Fourier transform of height data z(x).

The power spectral density of the Weierstass–Mandelbrot function follows apower law and is defined by

P(ω) = G2(D−1)

2 ln γω−(5−2D) (2.79)

Investigating the relationship between the power spectrum log P(ω) of the dis-crete data and log ω [22], the fractal dimensions can be deduced. The parametersG and D are found by using a log–log plot of the power spectrum of the fractal


0.6

0.3

0.0

0 1

Fractal simulation

2 3 4

−0.3

−0.6

0.6

0.3

0.0

0 1

Specimen t

Horizontal distance, x × 103 (m)

Horizontal distance, x × 103 (m)

Hei

ght,

y ×

106

(m)

Hei

ght,

y ×

106

(m)

2 3 4

−0.3

−0.6

2.0

1.0

0.0

0 2

Fractal simulation

(b)

(a)

4 6 8 10

−1.0

−2.0

2.0

1.0

0.0

0 2

Specimen s

4 6 8 10

−1.0

−2.0

Figure 2.20. Comparison between surface profiles and fractal simulations for two differ-ent specimen of stainless steel [36].

surface. If there is a linear log–log plot, the fractal dimension can be calculated.Indeed:

log(P (ω)) = log(G2(D−1)(2 ln γ )) + (2D − 5) log(ω) (2.80)

where D is thus deduced from the slope β and G is found from the intersectionof the vertical axis K [41]:

D = β + 5

2(2.81)

G = (10K × 2 ln γ )1/2(D−1) (2.82)


log [P (w)]

log (w)

K = log (G2(D−1)*2 In(g))

b = 2D−5

Figure 2.21. Log–log plot of the power spectral density function and deduction of fractalparameters.

By plotting the power spectrum density (Fig. 2.21), we can determine thefinite range of length scales for which the surface is fractal. At the spatial fre-quency, which the surface deviates from the power law behavior, the limits aredetermined. Mandelbrot [36] characterizes the two spectral regions of stainlesssteel specimens by splitting the Weierstrass–Mandelbrot function into two parts.

Practically, surface characterization and extraction of fractal parameters areachieved following this process: The surface is scanned in order to get surfaceheight information in the form of points (X, Z), the signal is treated with a fastFourier transform resulting in the power spectral density function that can beplotted, and the fractal characteristics are extracted from this plot.

2.4.5.2 Structure Function MethodA discrete form of the structure function of a profile z(x) is given by

S(τ) = 1

N − k

N−k∑i=0

(zi+k − zi)2 (2.83)

where k ranges from 1 to N . Defining τ as τ = 2π(xk − x1), the fractal structurefunction [23] of the Weierstrass function follows a power law that is given by[18, 68]

S(τ) = Cτ 4−2D where C = �(2D − 3) sin[(2D − 3)π/2]

(4 − 2D) ln γG2(D−1) (2.84)

where � is the gamma function with argument 2D − 3:

�(2D − 3) =∫ ∞

0t (2D − 2) exp−t dt (2.85)


The plot of S (τ ) as a function of τ (as for the power spectrum method) is astraight line in the log–log plot (Fig. 2.22):

log(S(τ)) = log(C) + (4 − 2D)︸︷︷︸β

log(τ ) (2.86)

If the slope of this line β satisfies 0 < β < 2, the profile is fractals and thefractals are found using the slope β and the intercept with the vertical axis K :

D = 4 − β

2(2.87)

G ={

10K (4 − 2D) ln γ

�(2D − 3) sin[(2D − 3)π/2]

}1/2D−2

(2.88)

In practice, the principle is the same as for the so-called power spectral density(PSD) function except that in this case the structure function is plotted.

2.4.5.3 Other Proposed MethodsOther methods have been proposed:

1. The cover method is an original approach. Let r be the yardstick and N(r)the repetitive measurement times each r . Suppose that N(r)×r is the lengthof the profile. For different r values, if there is a relationship between N(r)and (r), the fractal dimension is expressed by [22]

N(r) ∝ r−D (2.89)

2. The variation method uses the notion of δ variable to measure the variationof the profile in a δ neighborhood [17]. The order of growth of the δ variableis directly related to the fractal dimension of the profile.

3. The reticular cell counting method proposed by Gagnepain and Roques-Carmes [17] involves an iteration operation to an initial square whose areais supposed to be 1 and covering the entire graph. The method consists ofdividing the initial square into four subsquares and then each subsquare intofour subsquares, and so on. After n iterations, the initial square contains22n subsquares. The number of subsquares containing the discrete profileare counted and the length L of the profile is approximately obrained.The calculating equation of the fractal dimension D from this method isthen [22]

D = 1 + log L

n × log 2(2.90)


K = log (Γ(2D−3)sin[(2D−3)*π/2] / ((4−2D)*In(g))*G2(D -1))

b = 4−2D

log (S(t))

log [S(t)]

Figure 2.22. Log–log plot of the structure function and deduction of fractal parameters.

4. A modified Gaussian fractal model has been proposed by Zhou et al.[67] to derive equations to relate the bearing area curve with the fractaldimension D and the topothesy.

5. The difference average law (DAL) proposed by Jahn and Truckenbrodt[29] is based on an unusual power law for stochastically self-affine fractals.The authors state that the DAL procedure provides nearly the same relativefractal dimensions as the well-known methods (such as power spectrum)with smaller computation efforts.

2.4.5.4 ExampleThe idea is here to verify the feasibility of the power spectral density and structurefunction method:

1. A Weierstrass–Mandelbrot (WM) function is generated with fractalparameter D = 1.55 and amplitude parameter G = 1e−12. Points of theprofile z(x) are stored and z(x) is plotted in Figure 2.23(a) in dottedline.

2. The theoretical power spectral density and structure function are calcu-lated from Eq. 2.84 and Eq. 2.84 and plotted in log–log, respectively, inFigures 2.23(b) and 2.23(c) in dotted lines.

3. From points of the profile z(x), real power spectral density andstructure functions are calculated and plotted in log–log, respectively, inFigures 2.23(b) and 2.23(c) in plain lines. The intersection with axis andslopes allows one to find fractal parameters


0 0.2 0.4 0.6 0.8 1× 10−3

× 10−8

−5

−4

−3

−2

−1

0

1

2

3

4

Starting profile with D = 1.55 and G = 1e−12

Profile with power spectral density extractionProfile with structure function extraction

3 3.5 4 4.5 5 5.5 6−27

−26

−25

−24

−23

−22

−21

−20

−19

−18

(a)

(b)

log(

P(w

))

Real power spectral densityTheoretical power spectral density

log(w)

−5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2−17.5

−17

−16.5

−16

−15.5

−15

−14.5

log(t)

log(

S(t

))

Real structure functionTheoretical structure function

(c)

Figure 2.23. (a) Generated profile, (b) log–log plot of the power spectral density function,and (c) log–log plot of the structure function.


0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

4

meter meter

met

er

(a) D = 1.1

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

(b) D = 1.2

0 0.2 0.4 0.6 0.8 1−4−3−2−10123456

meter

met

er

(c) D = 1.3

0 0.2 0.4 0.6 0.8 1−8−6−4−202468

meter

met

er

(d) D = 1.4

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

meter

met

erm

eter

(e) D = 1.5

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

meter

met

er

(f ) D = 1.6

0 0.2 0.4 0.6 0.8 1−2

−1.5−1

−0.50

0.51

1.52

meter

met

er

(g) D = 1.7

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

4

meter

met

er

(h) D = 1.8

0 0.2 0.4 0.6 0.8 1

−5−4−3−2−1012345

meter

met

er

(i) D = 1.9

× 10−5 × 10−6× 10−4

× 10−7

× 10−9 × 10−9 × 10−11

× 10−7 × 10−8

× 10−3 × 10−3× 10−3

× 10−3

× 10−3 × 10−3

× 10−3 × 10−3

Figure 2.24. Weierstrass–Mandelbrot profile of 1 nm length, G = 1e−12 and differentvalues of D.

4. Finally the Weierstrass–Mandelbrot function is generated again with thenew parameters and plotted in Figure 2.23(a) for comparison with the initialprofile.

From the power spectral density function, the obtained parameters are D =1.32 and G = 2.39e−18. From the structure function, the obtained parameters areD = 1.56 and G = 1.31e−12. This illustrates the fact that the structure functionis a more accurate method for extracting fractal parameters of surfaces even ifboth methods are reliable.

2.4.5.5 Validity Domain for PSD and Structure Function MethodsDepending on the value of D, a method may be preferable to the other. It isthe aim of this section to investigate this. A Weierstrass–Mandelbrot function isgenerated with a 1-mm length and G = 1e−12 using the injected fractal parameterD (Fig. 2.24). Power spectrum and structure function are then plotted in order toextract the parameters. Extracted values are compared to injected values in order


1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90

5

10

15

20

25

30

Fractal parameter D

Err

or (

%)

D from PSD methodD from structure function method

Figure 2.25. Errors on fractal parameter D evaluation depending on the extractionmethod.

to get the error on the extracted parameters. Results are plotted in Figure 2.25.The power spectral density method gives better results than the structure functionfor values of D less than 1.3 even if both methods stay reliable in this interval.For all other values the structure function method has a greater efficiency.

Power function methods and structure function method are the most widelyused methods to:

1. Determine if a surface exhibits fractal properties.2. Extract the fractal parameters from the profile.

The power spectrum function is calculated by transforming the discrete heightsinto the frequency domain. This results in an approximation. Structure functionis calculated directly from the height information and results in a better approx-imation [18].

2.4.6 Conclusion

Different models have been reviewed. Most of them, however, use statisticalparameters in their representation. Fractal representation has been chosen becausefractal parameters do not depend on the measurement process and may be appli-cable to many engineered surfaces. The fractal character of surface finish forsome microfabrication processes has also been demonstrated. Extraction of fractalparameters can be done with the so-called structure function method.


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